The vanishing-off subgroup

In this paper, we define the vanishing-off subgroup of a nonabelian group. We study the structure of the quotient of this subgroup and a central series obtained from this subgroup.     

The nilpotent π-length of finite π-solvable groups with restrictions on subgroups

V.S. Monakhov’s conjecture concerning an estimate of the nilpotent π-length of a π-solvable group G is confirmed in the case l _p (G) ≤ 1 for all p ∈ π ? {q} and q ∈ π. New estimates of the nilpotent π-length of a π-solvable group with a supersolvable π-Hall subgroup are also obtained.     

The nilpotence class of the Frattini subgroup

Ap-group of sufficiently large nilpotence class cannot occur as a normal subgroup contained in the Frattini subgroup of any finite group. The Frattini subgroup of a group of order Π _pi ^αi with max α _i at least 3, has nilpotence class at most 1/2 (max α _i − 1). The Frattini subgroup of at-group is abelian. The occurrence of groups of orderp ⁴ as normal subgroups contained in Frattini subgroups is investigated. National Science Foundation Science Faculty Fellow, University of Cincinnati     

Detecting the index of a subgroup in the subgroup lattice

A theorem by Zacher and Rips states that the finiteness of the index of a subgroup can be described in terms of purely lattice-theoretic concepts. On the other hand, it is clear that if  is a group and is a subgroup of finite index of , the index cannot be recognized in the lattice of all subgroups of , as for instance all groups of prime order have isomorphic subgroup lattices. The aim of this paper is to give a lattice-theoretic characterization of the number of prime factors (with multiplicity) of .

     

Halving for the 2-Sylow subgroup of genus 2 curves over binary fields

We give a deterministic polynomial time algorithm to find the structure of the 2-Sylow subgroup of the Jacobian of a genus 2 curve over a finite field of characteristic 2. Our procedure starts with the points of order 2 and then performs a chain of successive halvings while such an operation makes sense. The stopping condition is triggered when certain polynomials fail to have roots in the base field, as previously shown by I. Kitamura, M. Katagi and T. Takagi. The structure of our algorithm is similar to the already known case of genus 1 and odd characteristic.     

The normal subgroup lattice of 2-transitive automorphism groups of linearly ordered sets

Using combinatorial and model-theoretic means, we examine the structure of normal subgroup lattices N(A()) of 2-transitive automorphism groups A() of infinite linearly ordered sets (, ). Certain natural sublattices of N(A()) are shown to be Stone algebras, and several first order properties of their dense and dually dense elements are characterized within the Dedekind-completion of (, ). As a consequence, A() has either precisely 5 or at least 2²¹ (even maximal) normal subgroups, and various other group- and lattice-theoretic results follow.     

The normal subgroup structure of ZM-groups

The main goal of this note is to determine and to count the normal subgroups of a ZM-group. We also indicate some necessary and sufficient conditions such that the normal subgroups of a ZM-group form a chain.